Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

Explicit descent in the Picard group of a cyclic cover of the projective line

Brendan Creutz

Abstract: Given a curve X of the form y^p = h(x) over a number field, one can use descents to obtain explicit bounds on the Mordell--Weil rank of the Jacobian or to prove that the curve has no rational points. We show how, having performed such a descent, one can easily obtain additional information which may rule out the existence of rational divisors on X of degree prime to p. This can yield sharper bounds on the Mordell--Weil rank by demonstrating the existence of nontrivial elements in the Shafarevich--Tate group. As an example we compute the Mordell--Weil rank of the Jacobian of a genus 4 curve over Q by determining that the$3-primary part of the Shafarevich--Tate group is isomorphic to Z/3 x Z/3.

Files available: paper (PDF)